Forecasting Thermoacoustic Instabilities in Liquid

Propellant Rocket Engines Using Multimodal Bayesian

Deep Learning

Ushnish Senguptaa, Gunther Waxenegger-Wilfingb, Jan Martinb,Justin Hardib, Matthew P. Junipera,∗

aDepartment of Engineering, University ofCambridge, Cambridge, Cambridgeshire, CB2 1PZ, United Kingdom

bInstitute of Space Propulsion, German Aerospace Center(DLR), Hardthausen, Baden-Wurttemberg, 74239, Germany

Abstract

The 100 MW cryogenic liquid oxygen/hydrogen multi-injector combustorBKD operated by the DLR Institute of Space Propulsion is a research plat-form that allows the study of thermoacoustic instabilities under realistic con-ditions, representative of small upper stage rocket engines. We use data fromBKD experimental campaigns in which the static chamber pressure and fuel-oxidizer ratio are varied such that the first tangential mode of the combustoris excited under some conditions. We train an autoregressive Bayesian neu-ral network model to forecast the amplitude of the dynamic pressure timeseries, inputting multiple sensor measurements (injector pressure/ tempera-ture measurements, static chamber pressure, high-frequency dynamic pres-sure measurements, high-frequency OH* chemiluminescence measurements)and future flow rate control signals. The Bayesian nature of our algorithmsallows us to work with a dataset whose size is restricted by the expense ofeach experimental run, without making overconfident extrapolations. Wefind that the networks are able to accurately forecast the evolution of thepressure amplitude and anticipate instability events on unseen experimentalruns 500 milliseconds in advance. We compare the predictive accuracy ofmultiple models using different combinations of sensor inputs. We find thatthe high-frequency dynamic pressure signal is particularly informative. We

∗Corresponding author. Email address: mpj1001@cam.ac.uk.

Preprint submitted to Acta Astronautica August 3, 2021

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also use the technique of integrated gradients to interpret the influence ofdifferent sensor inputs on the model prediction. The negative log-likelihoodof data points in the test dataset indicates that predictive uncertainties arewell-characterized by our Bayesian model and simulating a sensor failureevent results as expected in a dramatic increase in the epistemic componentof the uncertainty.

Keywords: thermoacoustic instabilities, multimodal data, bayesianmachine learning, interpretable machine learningPACS: 0000, 11112000 MSC: 0000, 1111

1. Introduction

High-frequency thermoacoustic instabilities are a potentially catastrophicphenomenon in rocket and aircraft engines. They usually arise from thecoupling between unsteady heat release rate and acoustic eigenmodes of thecombustion chamber. Acoustic waves incident on a flame cause fluctuationsin its heat release rate. If a flame releases more (less) heat than averageduring instants of higher (lower) local pressure, then more work is done bythe gas during the acoustic expansion phase than is done on it during theacoustic compression phase. If this work is not dissipated then the oscillationamplitude grows and the system becomes thermoacoustically unstable. Thiswas formalized by Chu (Equation 26 in [1]) who formulated the followingcriterion for instability:∫ τ

0

∫ V

0

p′(x, t)q′(x, t)dxdt >p0γ

γ − 1

∫ τ

0

∫ V

0

φ(x, t)dxdt (1)

where p′ and q′, are the perturbations in pressure and the heat release, re-spectively, p0 is the pressure of the unperturbed system, τ , V and φ arethe period of oscillation, the combustor volume and the viscous dissipationfunction, respectively.

Thermoacoustic instabilities can be challenging to model and predict be-cause they are usually sensitive to small changes in a combustion system[2]. The large pressure fluctuations and accelerated heat transfer stemmingfrom these oscillations can lead to structural failure, because rocket engineshave high energy densities and low factors of safety in their structural de-sign. From the F1 rocket engine, which underwent costly redesigns in the

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1960s [3] to the modern Japanese LE-9 engine [4], they continue to plaguethe development of liquid propellant rocket engines.

Designers can suppress thermoacoustic oscillations in three major ways.The first approach is to fit passive devices such as acoustic liners [3], baf-fles [5] or resonators [6]. The second is tuned passive control, where sensormeasurements are used to diagnose the onset of thermoacoustic instabilitieswhich can then be avoided by adjusting a passive device or the operatingpoint [7]. The third is active feedback control [8]. The challenging demandsof high power output, reliability and robustness placed on an active feedbackcontrol system means that feedback control is not used in rocket engines.

The construction of instability precursors from pressure measurementsand optical measurements has attracted considerable attention from the com-bustion research community [9]. Lieuwen [10] used the autocorrelation decayof combustion noise, filtered around an acoustic eigenfrequency, to obtain aneffective damping coefficient for the corresponding instability mode. Otherresearchers have used tools from nonlinear time series analysis, which capturethe transition from the chaotic behaviour displayed by stable turbulent com-bustors to the deterministic acoustics during instability, such as Gottwald’s0-1 test [11] and the Wayland test for nonlinear determinism [12] to assessthe stability margin of a combustor. Nair et al [13] reported that instabilityis often presaged by intermittent bursts of high-amplitude oscillations andused recurrence quantification analysis (RQA) to detect these. In a laterpaper [14], Nair and coworkers noted that combustion noise tends to lose itsmultifractality as the system transitions to instability. They proposed theHurst exponent as an indicator of impending instability. Measures derivedfrom symbolic time series analysis (STSA) [15] and complex networks [16]are also able to capture the onset of instability.

Recent studies have explored machine learning techniques to learn precur-sors of instability from data. These promise greater accuracy than physics-based precursors of instability, though at the cost of robustness and generaliz-ability. Hidden Markov models constructed from the output of STSA [17] ordirectly from pressure measurements [18] have been used to classify the stateof combustors. Hachijo et al [19] have projected pressure time series onto theentropy-complexity plane and used support vector machines (SVMs) to pre-dict thermoacoustic instability. SVMs were also employed by Kobayashi et al[20], who used them in combination with principal component analysis andordinal pattern transition networks to build precursors from simultaneouspressure and chemiluminiscence measurements. Sengupta et al [21] showed

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that the power spectrum of the noise can be used to predict the linear stabil-ity of a thermoacoustic eigenmode using Bayesian neural networks. Relatedwork by McCartney et al [22] used the detrended fluctuation analysis (DFA)spectrum of the pressure signal as input to a random forest and finds thatthis approach compares favorably to precursors from the literature. A recentstudy by Gangopadhyay and co-workers [23] trained a 3D Convolutional Se-lective Autoencoder on flame videos to detect instabilities. Transfer learningis also being explored [24] as a means of efficiently transferring knowledge ofprecursors across machines.

Most papers in the literature train indicators of approaching instability.In this study, on the other hand, we use nonlinear autoregressive time seriesmodeling with a Bayesian Neural Network to forecast, with uncertainties, thefuture amplitude of pressure fluctuations given the history of sensor signalsand future flow rate control signals. This informs the engine controller of theexpected timing and potential severity of an instability event. This approachalso lets us take advantage of the rich instrumentation of our rig by integrat-ing multimodal sensor data into our model: high frequency dynamic pressure(p′(t)) and OH* chemiluminescence (I ′(t)) measurements, injector pressures(pH2, pO2) and temperatures (TH2, TO2), chamber static pressure (pcc) andfuel flow rates (mO2, mH2). We compare models with different combinationsof sensors, different sensor-derived features and different lengths of signalhistories in the inputs. We find that, although operating parameters andmass flow rate control signals can anticipate instabilities on their own, inclu-sion of dynamic pressure signal history, in particular, results in a measurableincrease in the forecast accuracy. It is not possible to predict triggered ther-moacoustic instabilities from operating parameters and control signals alone,so the need to integrate multimodal sensor data with operating parametersand control signals in an instability forecasting framework is exigent.

The size of our dataset is limited by the high cost and effort requiredfor each additional test run on the rocket thrust chamber. Fortunately, theuncertainty-aware nature of the Bayesian Neural Network allows us to workin this medium-data regime without overconfident extrapolation. We com-pute log-likelihoods on the test dataset and find that the uncertainty is well-quantified. We also simulate a sensor failure event to assess the robustnessof our Bayesian forecasting tool. As expected, the network exhibits largeepistemic uncertainties during the failure event, signalling the unfamiliarout-of-distribution nature of the inputs. Finally we use the interpretationtechnique of integrated gradients (IG) to understand how the different input

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features in our model influence the predictions.

2. Experimental Setup

The dataset used in this study is derived from experiments performed onthe research combustor BKD [25] [26] operated at the P8 test facility [27][28] of the DLR Institute of Space Propulsion in Lampoldshausen. It hasthree main components: an injector head, a cylindrical combustion chamberand a convergent-divergent nozzle, as shown in Figure 1. The combustionchamber is water-cooled and is designed to deal with the high thermal loadsthat are expected during instability events. The L42 injector head has 42shear coaxial injectors and is operated with a liquid oxygen (LOX)/ hydrogen(LH2) propellant combination. The cylindrical combustion chamber is 80mm in diameter and the nozzle throat diameter is 50 mm, resulting in acontraction ratio of 2.56. The chamber static pressure (pcc) is varied between50−80 bar and mixture ratio of oxidizer to fuel (ROF = mO2/mH2) between2 − 6. The experiments considered here have a LOX injection temperatureTO2 around 110 K and a hydrogen injection temperature TH2 around 100 K.

For the operating point with pcc = 80 bar, ROF = 6, the total propellantmass flow rate is 6.7 kg/s, the theoretical thermal power is 100 MW and thethrust achieved is about 24 kN. These specifications place BKD at the lowerend of small upper stage engines.

A representative BKD test sequence, along with a spectrogram of dynamicpressure oscillations inside the combustion chamber, is shown in Figure 2.Stable and unstable operating conditions can be identified in the spectro-gram. Strong high-frequency combustion instabilities of the first tangential(1T) mode at about 10 kHz were excited consistently when the operatingpoint at pcc = 80 bar, ROF = 6 was approached. The coupling mecha-nism for this instability is injection-driven [29]. The flame dynamics aremodulated by the LOX post acoustics and combustion instabilities emergewhen the frequency of the 1T chamber mode matches the second longitudi-nal eigenmodes of the LOX posts. This mechanism has been confirmed usinghigh-speed flame imaging [30].

The extreme conditions within the thrust chamber, where temperaturesreach 3600 K and the pressure reaches 80 bar, limit the diagnostic instru-ments for our instability investigations. A specially designed measurementring is placed between the injector head and the cylindrical combustion

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Figure 1: Experimental thrust chamber BKD.

chamber segment, as shown in Figure 1. At this location, the tempera-tures are moderated by the injection of cryogenic propellants, which allowsthe mounted sensors to survive several test runs. Eight Kistler type 6043Awater-cooled high-frequency piezoelectric pressure sensors are flush-mountedin the ring with an even circumferential distribution, in order to measurethe chamber pressure oscillations p′(t). The high-frequency pressure sensorshave a measurement range set to ±30 bar and a sampling rate of 100 kHz.An anti-aliasing filter with a cutoff frequency of 30 kHz is applied. Threefibre-optical probes are used to record the OH* radiation intensity I ′(t) of theindividual flames. A small sapphire rod in the probe creates optical access.The full acceptance angle of the optical probes is approximately 2. In orderto capture the OH* radiation signal, the probes are equipped with inteferencefilters with a center wavelength of 310 nm. The sampling frequency of theI ′(t) signals is also 100 kHz. There are also several low sampling frequency(100 Hz) sensors that measure the chamber static pressure pcc, the mass flowrates of fuel mH2 and oxidizer mO2 and injector temperatures TH2, TO2.

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Figure 2: BKD test sequence and p′ spectrogram showing self-excited instability ofthe first tangential mode [29]. pcc is the chamber static pressure, ROF is the mixtureratio of oxidizer to fuel, and TH2, TO2 are the LOX and hydrogen injection temperaturesrespectively.

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3. Methods

3.1. Nonlinear autoregressive models with exogenous variables (NARX) fortimeseries modeling

To forecast instabilities, we train NARX-style models [31] to capture thefunctional relationship between the history of the sensor data, control sig-nals and future pressure fluctuations. These model the evolution of pressurefluctuations as follows:

log(p′rms(tm + ∆t)) = f(xm,xm−1, ...xm−h,um+f ) + σε (2)

Here, p′rms(tm + ∆t) is the root mean square amplitude of the dynamicpressure fluctuation signal in a 50 ms window centered around tm + ∆t. Wechoose ∆t = 500 ms for our models, because we consider this to be sufficientwarning for a rocket engine controller to react and take appropriate action.The state variables xm−i consist of various sensor data (or transformationsthereof) at times tm− 50× i ms. The order of dependence h determines howfar in the past we look. In this study, we explore models with h = 0 andh = 2. The elements of um+f are the two mass flux control signals mH2,set

and mO2,set averaged over the window between tm and tm + ∆t. These arethe exogenous variables whose future values are known and can therefore beincluded in the inputs. It is possible to train models that capture higherorder dependence on instantaneous values of the control signals instead ofreducing them to an average, but we find that this does not improve predic-tive accuracy in our problem. σε is the homoskedastic (constant) observationnoise or aleatoric uncertainty term, which is assumed to be Gaussian. Weforecast the logarithm of p′rms(tm + ∆t) instead of p′rms(tm + ∆t) because thenoise in the pressure fluctuations increases with the amplitude, and so thesimplifying assumption of constant homoskedastic noise only holds approxi-mately once this logarithmic transform has been applied. It also forces p′rms

predictions to be positive.We model the nonlinear function f in Equation (2) as a feedforward neural

network with a simple multi-layer perceptron (MLP) architecture. In a stan-dard neural network, one obtains a point estimate of the network parameters.Here, however, we train a machine learning algorithm on a dataset whose sizeis limited by the cost of performing experiments on large rocket thrust cham-bers. With a standard neural network this size limit could lead to naive andinaccurate extrapolations when the network encounters out-of-distribution

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inputs that are different from those which were experienced during train-ing. In safety-critical systems such as rockets, control decisions based onoverconfident predictions could result in dire consequences. Bayesian Neu-ral Networks (BNNs) are an elegant solution to this problem because theyprovide a confidence interval as well as a prediction.

3.2. Bayesian Neural Network Ensembles

The Bayesian framework for training neural networks [32] involves plac-ing a sensible prior probability distribution p(θ) over the parameters θ ofthe network. Observations D are then used to update our belief about theparameter distribution. The posterior distribution p(θ|D) is obtained usingBayes’ rule.

p(θ|D) =p(D|θ)p(θ)∫p(D|θ)p(θ)dθ

(3)

Bayesian models are resistant to overfitting on small datasets and un-certainty aware. They are also useful for continual learning throughout thelifetime of a device in operation [33]. Exact Bayesian inference, however, isnot possible in neural network models because the integral

∫p(D|θ)p(θ)dθ

is analytically intractable. The most widely used numerical method to in-tegrate over the posterior, Markov Chain Monte Carlo, is too computation-ally expensive to be practical. Variational inference is often used [34], inwhich the posterior is approximated using a convenient parametrization andthe Kullback-Leibler divergence between this variational distribution and thetrue posterior is minimized using backpropagation. However, although com-putationally cheap, mean-field variational inference also has drawbacks suchas not capturing correlations between parameters. Recently, a different ap-proximate inference method, based on ensembling, has been proposed. Thisis called the anchored ensembling algorithm. It is cheap, simple and scal-able but manages to outperform variational inference in several uncertaintyquantification benchmarks [35].

Consider a dataset of ND data points (xn, yn), where each data pointconsists of features xn ∈ RD and output yn ∈ R. Define the likelihood foreach data point as p(yn | θ,xn, σ2

ε ) = N (yn | NN(xn; θ), σ2ε ), where NN is a

neural network whose weights and biases form the latent variables θ while σ2ε

is the observation noise. Define the prior on the weights and biases θ to bethe standard normal p(θ) = N (θ | µprior,Σprior). The anchored ensemblingalgorithm then does the following:

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Figure 3: Approximate Bayesian inference using ensembles of neural networks.

1. The parameters θanc,j of each j-th member of our neural network ensem-ble are initialized by drawing from the prior distributionN (µprior,Σprior).

2. Each ensemble member is trained using a modified loss function thatanchors the parameters to their initial values. The loss function forthe j-th ensemble member is given by Lossj =

∑ND

i=1(yi − yj(xi, ti))2 +

‖Σ−1/2prior(θj − θanc,j)‖22, where yj(xi, ti) is the neural network output andthe i-th diagonal element of Σ is the ratio of data noise to the priorvariance of the i-th parameter.

Pearce et al [35] prove that this procedure approximates the true posteriordistribution for wide neural networks and that the trained neural networksin the ensemble may be treated as samples from an approximate posteriordistribution. The resulting ensemble has predictions that converge when theyare well-supported by the training data and diverge when they are not. Thevariance of the ensemble predictions is an estimate of the Bayesian model’sepistemic uncertainty. Epistemic or knowledge uncertainty is the reducibleportion of uncertainty which stems from uncertainty in model parameters anddecreases as more data is added. The total uncertainty of each prediction isobtained by adding the epistemic uncertainty and the irreducible aleatoricuncertainty σε which stems from observation noise:

σ2total = σ2

ε +1

Ne

∑j

y2j − (1

Ne

∑j

yj)2 (4)

3.3. Interpretation using Integrated Gradients

Neural networks have a reputation as black boxes, which disincentivizestheir application to cases in which practitioners must understand why the

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algorithm is or is not working. We use the technique of integrated gradients(IG) [36] to attribute the predictions of our network ensemble to the inputfeatures. This is a simple scalable method that only requires access to thegradient operation. IG has several desirable properties that other attribu-tion methods lack, such as implementation invariance, sensitivity, linearity,completeness and preservation of symmetry.

For deep neural networks, the gradients of the output with respect to theinput are an analog of the linear regression coefficients. In a linear model, theregression coefficients characterize the relationship between input and outputvariables globally. In a nonlinear neural network, however, the gradient ofthe output at a point merely characterizes the local relationship between apredictor variable and the output. IG computes the path integral of thegradient of the outputs with respect to the inputs around a baseline to theinput under consideration. For an image recognition algorithm, a completelyblack image could be a reasonable choice of baseline, while for a regressionproblem like ours, where the input variables were normalized to have zeromean and unit standard deviation, the average input of all zeros is a sensiblebaseline. We consider the straight-line path (in the feature space Rp) fromthe baseline x′ to the input x being considered. The IG attribution attrj forthe j-th feature is then defined as follows:

attrj(x) = (xj − x′j)

∫ 1

α=0

∂f(x′ + α(x− x′))

∂xjdα (5)

This integral is computed numerically by sampling the gradient evenly alongthe path. For our neural network ensemble, we average the integrated gra-dient attributions obtained from individual ensemble members to obtain amean attrj for each feature.

3.4. Transforming sensor signals into neural network inputs

We compare NARX models with different combinations and transfor-mations of the sensor data in their state variables xm−i (Equation 2) andevaluate their accuracy.

Models that include the low frequency signals pcc, mH2, mO2, TH2 and TO2

in their xm−i vectors use the average value of the signal between tm − 50iand tm − 50(i + 1). For the high frequency pressure und OH* signals, p′

and I ′, we use transformations from the literature that are known to cap-ture instability-relevant information as described next. The current p′rms is

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a natural candidate which is included in all models that incorporate pres-sure information. Sengupta et al [21] and McCartney et al [37] have shownthat before a combustor transitions to instability, the power spectral densityshows measurable changes which can be used as a precursor. We estimatethe spectral density of the p′ (p′FFT) and I ′ (I ′FFT) time series by applyingWelch’s method [38] to a segment of the signals between tm − 50i ms andtm − 50(i + 1) ms. Welch’s method divides the signal into overlapping seg-ments and applies the Fast Fourier Transform (FFT) before averaging theresult, leading to lower frequency resolution but a higher signal-to-noise ra-tio. In this study, a Hann window of length 30 with 50% overlap is used forboth p′ and I ′.

Kobayashi [20] applied machine learning to the the ordinal partition tran-sition network of the p′ and I ′ timeseries to detect instabilities early. Theordinal partition transition networks for an m-dimensional time series are ex-pressed by a weighted adjacency matrix W consisting of Wij = P (πi −→ πj),i, j ∈ [1, 2m],

∑Wij = 1, where P (πi −→ πj) is the observed probability from

the ith- to jth-order transition patterns. In this study, the order patternsare π1, π2, π3, and π4 which account for all combinations of the signs of theincrements for our two-dimensional time series [p′(t), I ′(t)]. The probabilitydistribution of the corresponding 16-dimensional transition patterns W com-puted from the segment of the signals between tm − 50i and tm − 50(i + 1)are used in the state variable vector.

We also include a naive baseline model p′rms(tm + ∆t) = p′rms(tm) forcomparison. This model, by design, is incapable of forecasting any futureinstabilities but, because large portions of the run consist of steady operation,its predictions have a reasonably good root mean squared error (RMSE). Thismodel exists to provide a baseline RMSE and any useful forecast needs to besignificantly more accurate than this.

3.5. Train-test-validation split, hyperparameters, performance metrics

We use a dataset of five experimental runs: three that are 50 secondslong and two that are 90 seconds long. Each run has two occurrences of the1T instability, where an instability event is defined as being when the peak-to-peak amplitude of the pressure fluctuations exceeds 5% of the chamberstatic pressure.

To avoid data leakage, one experimental run is exclusively reserved for hy-perparameter tuning. The timeseries is split into contiguous 250 millisecondblocks and every fifth block is used to evaluate the negative log-likelihood,

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which is the metric we use for tuning the BNN hyperparameters. The hyper-parameters are the network depth, layer width, noise σε, the learning rate forthe ADAM optimizer, the number of epochs and the number of neural net-works in the ensemble. For simplicity, and to facilitate comparison betweenmodels, the same hyperparameters are chosen for all models. This does notcompromise the performance of any particular model, since increasing thekey hyperparameters, width and depth, benefits all models. We choose a 3hidden layer MLP with 100 units in each layer because larger networks showonly minimal gains in performance. A learning rate of 2 × 10−4 is found tobe optimal for convergence. The aleatoric noise parameter σε is set to 0.07.In anchored ensembling, we train all members of the ensemble until conver-gence so the number of epochs was set to 512, which is found sufficient forall models. Converged estimates of the test data log-likelihood are obtainedusing 25 neural networks per ensemble.

To evaluate model performance, leave-one-out cross validation (LOOCV)is performed on the remaining four runs. Time series data involves strongtemporal correlations, so assigning training and test data points randomlyis dishonest because then the two sets would be highly similar. The test setmust therefore be a completely independent experimental run. Additionally,LOOCV is necessary because with only four runs, firm conclusions cannotbe drawn based on performance metrics for one particular test-train split,since any claimed efficacy or differences between models could be causedby chance. We report average performance metrics across the four possiblesplits.

To evaluate model accuracy, we compute the root mean squared error(RMSE) of our pressure amplitude forecast for the entire test run (fullRMSE). We also calculate this on a subset of the test run comprising thetwo segments of the timeseries one second before and after a transition toinstability event (transition RMSE). The RMSE is computed separately onthis subset because our primary interest is in forecasting instabilities andthe relative merits of different models become clearer on this subset wherethere is a dramatic change in the amplitude. As a measure of the quality ofuncertainty, we report mean negative log-likelihood per test data point.

4. Results

The models were trained on a laptop with 16 GB RAM, an Intel i7-10870HCPU and an NVIDIA RTX 2070 GPU. Each NARX model took ∼ 2.5 hours

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to train. The inference time of the ensemble averaged around 10 millisecondsper sample, indicating that the algorithm is suitable as a real-time diagnostictool.

Table 1 shows the average prediction RMSEs for the NARX models com-puted during cross-validation across the four runs. We note, firstly, that theNARX models outperform the naive baseline, which means that the datacontains a predictable signal that can be extracted. We also observe that thesimple models, which use only the two operating point parameters pcc andROF as inputs, perform well. This is unsurprising because we know thatthe 1T instabilities are consistently triggered when the operating point atpcc = 80 bar and ROF = 6 is approached. This means that knowing the cur-rent operating point and future flow rate control signals is informative. Alltransitions to instability in the test runs differ from each other, however, eventhough though they have identical control trajectories. The cross-validationshows unambiguously that the data from the additional instrumentation im-proves the prediction of future pressure fluctuations. Models that include thedynamic pressure spectrum p′FFT, radiation intensity spectrum I ′FFT, currentamplitude p′rms, ordinal network transition probabilities, W , TO2, TH2, pO2

or pH2, have better prediction accuracies, especially on the transition subset.Including higher order dependencies on state variables also provides a slightboost in accuracy. Interestingly, the dynamic pressure spectrum appearsmore informative than the spectrum of the OH* radiation intensity. Thiscould be due to the fact that a single optical probe observes only a smallvolume of the chamber, while the pressure field integrates information acrossthe whole chamber.

Table 1 also reports the mean negative log-likelihood (NLL) per data-point, averaged across the four cross-validations. The mean negative log-likelihoods are fairly low for most of our models, implying that very fewobservations lie far outside the uncertainty bounds of the forecast (Figure4). This is also confirmed by the percentage of data points in the test setwithin the ±1 s.d. and ±3 s.d. bounds, which are roughly in line with ourGaussian assumptions (68% within 1 s.d., 99.7% within 3 s.d.). Additionally,though models using more features have higher prediction accuracies, theyhave slightly worse NLLs in some cases because the presence of additionalfeatures results in higher epistemic uncertainties.

To test the robustness of the Bayesian model against out-of-distributioninputs, we simulate a sensor failure event. This is shown in Figure 5. Tosimulate an optical probe icing event between 10 sec and 40 sec, the original

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Figure 4: Forecast p′rms values for the four experimental runs based on sensor data priorto t− 0.5 sec, superimposed on observed p′rms values at time t. The model was trained onthe other three runs in each case. This model used p′FFT and p′rms as state variables andorder h = 2. Actual observations are shown as black dots, 3 S.D. total uncertainty boundsfor the forecast are in light blue and 3 S.D. epistemic uncertainty bounds are in deep blue.

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Figure 5: Bayesian Neural Network epistemic uncertainty (top) and p′rms forecasts (bot-tom) for experimental run no. 4 where a fibre optical probe failure was simulated betweent = 10 sec and t = 40 sec (marked by two grey dotted lines). The model used I ′FFT, p

′rms

as input state variables and order = 0. Actual observations are shown as black dots, 3S.D. total uncertainty bounds for the forecast are represented by light blue bars and 3S.D. epistemic uncertainty bounds are in deep blue.

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State variable inputs x Order Full RMSE Transition RMSE NLL

Baseline – 0.1552 0.2134 –pcc, ROF 0 0.1289 0.1630 -1.622pcc, ROF 2 0.1202 0.1576 -1.631p′rms, p

′FFT 0 0.1197 0.1546 -1.5752

p′rms, p′FFT 2 0.1169 0.1508 -1.5467

p′rms, I′FFT 0 0.1255 0.1612 -1.5154

p′rms, W 0 0.1241 0.1566 -1.5801TO2, TH2, pO2, pH2 0 0.1243 0.1578 -1.5948

pcc, ROF, p′rms, p′FFT, TO2, TH2, pO2, pH2 0 0.1048 0.1407 -1.5632

pcc, ROF, p′rms, p′FFT, TO2, TH2, pO2, pH2 2 0.1005 0.1311 -1.5350

Table 1: Cross-validation RMSEs (in bar) for entire runs (Full RMSE) and the transitionsubset of the run (Transition RMSE) and mean negative log-likelihoods (NLL) for differentNARX models. Lower RMSEs and NLLs are better.

photomultiplier signal is replaced by white noise with mean 0.05 and ampli-tude 0.015. During this simulated probe failure, the epistemic uncertaintyof the Bayesian neural network greatly increases, indicating that the inputsare highly dissimilar to conditions experienced during training. Predictionsremain reasonable despite the uninformative optical signal, although theybecome much less accurate.

We use the technique of integrated gradients to produce feature-level at-tribution plots, which can tell us how much a particular predictor influencedthe prediction for a particular input. For example, the attribution plots inFigure 6 shows this technique being applied to two data points. The firstone is a data point just prior to the first instability event in a run, wherea large jump in the dynamic pressure fluctuation amplitude is forecast bythe model. This model uses p′FFT, p′rms and the flowrate control signals asinput. The most prominent feature is the large positive attribution for theoxygen flowrate control signal mO2, whose feature value is also large for thisdatapoint. This is expected, because a large mO2 would increase the pccand ROF, bringing the combustor closer to instability. The prediction isalso modulated by contributions from the p′FFT features. This datapoint hasa concentration of power around the 10 kHz spectrum and low power inthe mid to higher frequencies. There is a positive contribution from the 10kHz frequency component of the spectrum, which is the frequency of the 1T

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Figure 6: Integrated gradient attributions and input feature values (normalized units)for two datapoints from experimental run no. 4 at t = 16.0 seconds (top, immediatelypreceding an instability) and t = 60.0 seconds (bottom, stable). The model used p′FFTand p′rms as state variables and order = 0. The blue dots show the normalized values ofthe features for this particular datapoint and the bars represent the integrated gradientattributions for each feature. For example, feature 6 (spectral power around the 16.66 kHzfrequency) in the top plot has a large negative value and a positive integrated gradientsattribution, meaning that the network has learned that low power in this frequency bandis a potential indicator of instability.

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mode, as well as from the 16.66k Hz frequency. This shows that the modelhas learned that an increased concentration of acoustic power around thefrequency of the 1T mode and away from the mid-frequencies is an indica-tion that the system is close to instability. The second data point is fromstable operation. The attributions in this case have mostly negative valuesbecause the prediction of the model is strongly negative. The current pres-sure fluctuation amplitude p′rms is close to −1.0 in this case and naturally hasa large negative attribution. The mH2 input (here −0.78) also contributesnegatively because low ROFs are stable. Finally, the high concentration ofpower in the mid-frequencies also tells the network that this operating pointwill stay stable.

5. Conclusions

In this paper, we present the application of an uncertainty-aware andinterpretable machine learning algorithm to forecast instabilities using mul-timodal sensor data in a realistic, thermoacoustically unstable rocket thrustchamber. We predict the amplitude of dynamic pressure fluctuations 500 msin advance using temperature, static pressure, high-frequency dynamic pres-sure and OH* radiation intensity data recorded during test runs performedwith the cryogenic rocket thrust chamber BKD. We perform a rigorous cross-validation of the autoregressive Bayesian neural network and find that ourmodels are able to anticipate all the instability events with varying accu-racy (25−40% lower RMSEs on the transition subset compared to baseline),depending on the combination of inputs chosen. The inclusion of longer histo-ries and additional sensor information, in general, boosts predictive accuracy.Models that use the dynamic pressure signal, in particular, show a measur-able improvement in the forecast, confirming the findings of the literature onthis topic.

A key benefit of our Bayesian approach is the predictive uncertainty.In safety-critical machines such as rocket engines, a diagnostic tool mustbe robust to unexpected events such as sensor failures. We compute thelog-likelihood of the test data given model predictions and find that uncer-tainties are well-characterized. We also simulate a sensor failure event in amodel whose inputs included the spectrum of the fibre-optical probe signaland demonstrate the robustness of Bayesian algorithms. Integrated Gradi-ents helps us identify important features and how they influence a particularprediction of the neural network.

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Future work will focus on exploring the effectiveness of these tools onDLR’s other experimental datasets, such as the LOX-methane tests, whichalso displayed injection-coupled acoustic instabilities [39]. We will also ex-plore the generalizability of data-driven diagnostics across multiple datasetsand combustors, using approaches that utilise transfer learning or domaininvariance. Hardware upgrades are currently being undertaken at the testfacility that will allow the implementation of these methods into the enginecontrol algorithm [40].

6. Acknowledgements

The authors would like to thank the crew of the P8 test bench and alsoStefan Groning for preparing and performing the test runs. Furthermore,we thank Wolfgang Armbruster for helping us process and understand thedata. Ushnish Sengupta is an Early Stage Researcher within the MAGISTERconsortium which receives funding from the European Union’s Horizon 2020research and innovation programme under the Marie Sk lodowska-Curie grantagreement No 766264.

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